US6874369B2 - Stress measurement method using X-ray diffraction - Google Patents

Stress measurement method using X-ray diffraction Download PDF

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US6874369B2
US6874369B2 US10/650,059 US65005903A US6874369B2 US 6874369 B2 US6874369 B2 US 6874369B2 US 65005903 A US65005903 A US 65005903A US 6874369 B2 US6874369 B2 US 6874369B2
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ray
specimen
stress
strain
hkl
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US20040177700A1 (en
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Ryouichi Yokoyama
Kamihisa Endo
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Rigaku Corp
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N23/00—Investigating or analysing materials by the use of wave or particle radiation, e.g. X-rays or neutrons, not covered by groups G01N3/00 – G01N17/00, G01N21/00 or G01N22/00
    • G01N23/20—Investigating or analysing materials by the use of wave or particle radiation, e.g. X-rays or neutrons, not covered by groups G01N3/00 – G01N17/00, G01N21/00 or G01N22/00 by using diffraction of the radiation by the materials, e.g. for investigating crystal structure; by using scattering of the radiation by the materials, e.g. for investigating non-crystalline materials; by using reflection of the radiation by the materials
    • G01N23/207—Diffractometry using detectors, e.g. using a probe in a central position and one or more displaceable detectors in circumferential positions
    • H—ELECTRICITY
    • H10—SEMICONDUCTOR DEVICES; ELECTRIC SOLID-STATE DEVICES NOT OTHERWISE PROVIDED FOR
    • H10P—GENERIC PROCESSES OR APPARATUS FOR THE MANUFACTURE OR TREATMENT OF DEVICES COVERED BY CLASS H10
    • H10P14/00—Formation of materials, e.g. in the shape of layers or pillars
    • H10P14/60—Formation of materials, e.g. in the shape of layers or pillars of insulating materials
    • H10P14/69—Inorganic materials
    • H10P14/692—Inorganic materials composed of oxides, glassy oxides or oxide-based glasses
    • H10P14/6938—Inorganic materials composed of oxides, glassy oxides or oxide-based glasses the material containing at least one metal element, e.g. metal oxides, metal oxynitrides or metal oxycarbides
    • H10P14/69398—Inorganic materials composed of oxides, glassy oxides or oxide-based glasses the material containing at least one metal element, e.g. metal oxides, metal oxynitrides or metal oxycarbides the material having a perovskite structure, e.g. BaTiO3

Definitions

  • the present invention relates to a stress measurement method using X-ray diffraction.
  • the present invention relates to a stress measurement method for a c-axis-oriented specimen of a tetragonal polycrystal.
  • a sin 2 ⁇ method has been used with for the stress measurement method using X-ray diffraction.
  • the sin 2 ⁇ method requires four conditions: (1) the crystal grain is small, (2) there is no strong preferred orientation, (3) a plane stress state is established within the depth of penetration of the X-ray, and (4) no stress gradient is present in the depth direction.
  • the stress measurement of a polycrystalline specimen, in which a specific crystal axis orients in a specific direction does not satisfy the above-described condition (2) including no strong preferred orientation. Therefore, the stress measurement of the fiber texture specimen using the sin 2 ⁇ method provides not a correct value but an approximate value.
  • a first aspect of the present invention is a measurement method, in which a stress of a c-axis-oriented specimen of a tetragonal polycrystal is measured using X-ray diffraction under the assumption of a plane stress state.
  • the target for the measurement is limited to the tetragonal system having the Laue symmetry 4/mmm.
  • the method according to this aspect comprises the steps of: (a) preparing a c-axis-oriented specimen of a tetragonal polycrystal as a specimen to be measured; (b) defining, as a specimen coordinate system, a coordinate axis P 3 perpendicular to a surface of the specimen and two coordinate axes P 1 and P 2 orthogonal to each other within the specimen surface, and arranging an X-ray optical system including an X-ray source and an X-ray detector within a plane including the coordinate axes P 1 and P 3 ; (c) selecting one set of Miller indices (hkl) of the specimen, and arranging the X-ray source and the X-ray detector to be symmetrical with a normal to a crystal plane with said one set of Miller indices (hkl), the normal to the crystal plane being inclined from a normal to the specimen surface at an angle of ⁇ , so that a diffracted X-ray from the crystal plane with said one set of Miller indices (h
  • a second aspect of the present invention is a measurement method, in which a stress of a c-axis-oriented specimen of a tetragonal polycrystal is measured using X-ray diffraction under the assumption of an equi-biaxial stress state.
  • the targets for the measurement are both of the tetragonal system having the Laue symmetry 4/mmm and that having the Laue symmetry 4/m.
  • the method according to this aspect comprises the steps of: (a) preparing a c-axis-oriented specimen of a tetragonal polycrystal as a specimen to be measured; (b) defining, as a specimen coordinate system, a coordinate axis P 3 perpendicular to a surface of the specimen and two coordinate axes P 1 and P 2 orthogonal to each other within the specimen surface, and arranging an X-ray optical system including an X-ray source and an X-ray detector within an arbitrary plane including the coordinate axis P 3 ; (c) selecting one set of Miller indices (hkl) of the specimen, and arranging the X-ray source and the X-ray detector to be symmetrical with a normal to a crystal plane with said one set of Miller indices (hkl), the normal to the crystal plane being inclined from a normal to the specimen surface at an angle of ⁇ , so that a diffracted X-ray from the crystal plane with said one set of Miller indices (hkl
  • Materials belonging to the tetragonal system having the Laue symmetry 4/mmm include BaTiO 3 , CuGaS 2 , MgF 2 , PbTiO 3 , Mn 3 O 4 , MnF 2 , MnO 2 , TiO 2 and YVO 4 .
  • Typical materials of the tetragonal system having the Laue symmetry 4/m include PbMoO 4 and CaWO 4 .
  • FIG. 1 is a perspective view for explaining a c-axis-oriented specimen of a tetragonal polycrystal
  • FIG. 2 shows formulae (1) to (4) representing the principle of a stress measurement using X-ray diffraction
  • FIG. 3 is a perspective view showing three coordinate systems which will be used for explaining a calculation method in a stress measurement
  • FIG. 4 is a diagram for explaining transformation matrices ⁇ , ⁇ and ⁇ between the three coordinate systems
  • FIG. 5 is a diagram showing a system of notation of an elastic compliance constant S, a stress ⁇ and a strain ⁇ in the three coordinate systems;
  • FIG. 6 is a diagram showing the relationship between an elastic compliance constant in tensor notation and a 6 ⁇ 6 matrix in matrix notation
  • FIG. 7 shows formulae (5) to (14) used for stress calculation
  • FIG. 8 is a diagram showing elastic compliance constants of two types of tetragonal system in matrix notation
  • FIG. 9 shows formulae (17) to (20) used for stress calculation
  • FIG. 10 shows formulae (21) and (22) used for stress calculation
  • FIG. 11 shows a formula (23) used for stress calculation
  • FIG. 12 shows formulae (24) and (25) used for stress calculation
  • FIG. 13 shows a formula (26) used for stress calculation
  • FIG. 14 shows formulae (27) to (30) used for stress calculation
  • FIG. 15 is a diagram showing the symmetrical property of the Laue symmetry 4/mmm
  • FIG. 16 is a table showing angles of eight equivalent crystal coordinate systems
  • FIG. 17 is a table showing the values of ⁇ , ⁇ , d 0 and ⁇ 0 with respect to each set of the Miller indices (hkl) of PbTiO 3 ;
  • FIG. 18 is a diagram for explaining an X-ray optical system.
  • FIG. 1 illustrating a c-axis-oriented specimen of a tetragonal polycrystal
  • most of many crystal grains which are present in the vicinity of the surface of a specimen 10 have the c-axes of the tetragonal system perpendicular to the specimen surface.
  • the remaining two axes (since these are equivalent to each other, both axes are referred to as a-axes) of the crystal axes of the tetragonal system are present within a plane parallel to the specimen surface.
  • directions of the two a-axes are random, and randomly oriented crystal grains are mixed with each other.
  • Such a c-axis-oriented specimen is the target for measurement of the present invention.
  • Formula (1) is the Bragg's equation representing the diffraction condition of X-ray. Under the assumption that the wavelength ⁇ (the wavelength of an X-ray used for the measurement) is constant, both sides of formula (1) are totally differentiated and transformed to become formula (2). On the other hand, the strain ⁇ is defined by formula (3). When the Bragg's diffraction angle of a crystal plane in a non-strain state is represented by ⁇ 0 , formula (4) is derived from formulae (2) and (3).
  • the strain ⁇ can be determined through measurement of the diffraction angle ⁇ using X-ray diffraction.
  • the above-described principle and the conventional sin 2 ⁇ method based thereon are described in detail in, for example, Jikkenbutsurigaku kouza 20, X-sen kaisetsu (Experimental physics course 20, X-ray diffraction) edited by Kohra, K., KYORITSU SHUPPAN CO., LTD., 1988, p. 571-575 “16.2X-sen ouryokusokuteino genri (Principle of X-ray stress measurement)”.
  • FIG. 3 is a perspective view showing three coordinate systems which will be used for explaining a calculation method in a stress measurement.
  • a specimen coordinate system P With respect to a specimen 10 having a flat surface, three kinds of three-dimensional orthogonal coordinate systems are taken into consideration: a specimen coordinate system P, a crystal coordinate system X and a laboratory coordinate system L.
  • the specimen coordinate system P is a three-dimensional orthogonal coordinate system fixed on the specimen. Two coordinate axes P 1 and P 2 orthogonal to each other are defined within the surface of the specimen, and the coordinate axis P 3 is set to become perpendicular to the specimen surface.
  • the specimen coordinate system P is visible to the observer, and becomes a standard coordinate system to the observer. The stress applied to the specimen is determined with this specimen coordinate system.
  • the crystal coordinate system X is a three-dimensional orthogonal coordinate system representing crystal axes of a crystal grain to which the crystal plane contributing to the diffraction belongs, the crystal grain being present in the vicinity of the specimen surface.
  • a fiber texture polycrystal is contemplated as the specimen and, therefore, a specific crystal axis (in this case, the coordinate axis X 3 ) of every crystal grain contained in the specimen is perpendicular to the specimen surface.
  • the other two coordinate axes X 1 and X 2 are present within the surface of the specimen. This crystal coordinate system X is not visible to the observer, and the coordinate axes X 1 and X 2 may point in a random direction.
  • the crystal coordinate system X is derived by rotation of the specimen coordinate system P around the coordinate axis P 3 in a counterclockwise direction through an angle of ⁇ when viewed from the origin of P 3 toward the forward of P 3 .
  • P 3 and X 3 coincide with each other.
  • the laboratory coordinate system L is based on an X-ray optical system in the X-ray diffraction measurement.
  • the laboratory coordinate system L is derived in a manner that the specimen coordinate system P is rotated around the coordinate axis P 3 in a clockwise direction through an angle of ⁇ when viewed from the origin of P 3 toward the forward of P 3 and, in addition, P 3 and P 1 are rotated around P 2 at that time (P 2 is present in the location of L 2 shown in FIG. 3 ) through an angle of ⁇ .
  • the direction of L 3 is the direction of the normal to the crystal plane contributing to the diffraction.
  • This laboratory coordinate system L is used for measuring the diffraction angle (that is, for measuring the strain).
  • FIG. 5 A system of notation of an elastic compliance constant S, a stress ⁇ and a strain ⁇ in each of the coordinate systems is shown in FIG. 5 .
  • these transformation matrices can be represented by formulae (5) to (7) shown in FIG. 7 .
  • These formulae can be represented using a rotation matrix R 1 ( ⁇ ) which rotates around a coordinate axis 1 through an angle of ⁇ , a rotation matrix R 2 ( ⁇ ) which rotates around a coordinate axis 2 through an angle of ⁇ and a rotation matrix R 3 ( ⁇ ) which rotates around a coordinate axis 3 through an angle of ⁇ .
  • the forms of the rotation matrices R 1 to R 3 are represented by formulae (8) to (10) as shown in FIG. 7 .
  • the tetragonal system can be classified based on the symmetry thereof, and be classified into the two types: the Laue symmetry (that is, the symmetry of the reciprocal lattice space) 4/mmm and that having the Laue symmetry 4/m.
  • the former includes a four-fold symmetry about the c-axis and three mirror symmetries and, therefore, has high symmetry.
  • the latter includes a four-fold symmetry about the c-axis and one mirror symmetry and, therefore, has low symmetry.
  • the symmetry of the crystal is described in, for example, X-sen kesshoukaisekino tebiki, Ouyoubutsurigaku sensho (Guide to X-ray crystal analysis, Applied physics sampler), Sakurai, T., Shokabou, 1983, p. 53.
  • the single-crystal elastic compliance constant S of the tetragonal system belonging to 4/mmm can be represented by formula (15) shown in FIG. 8 .
  • the single-crystal elastic compliance constant S of the tetragonal system belonging to 4/m can be represented by formula (16) shown in FIG. 8 .
  • each stress state will be ascertained whether the stress can be measured using X-ray diffraction, that is, whether the stress can be experimentally determined based on the relationship between the measured value of the strain ⁇ and the measurement conditions of ⁇ and the like.
  • the stress can be measured based on formula (19) shown in FIG. 9 with respect to both of the tetragonal system belonging to 4/mmm and 4/m. That is, when the strain ⁇ L 33 and sin 2 ⁇ are plotted on a graph, there is a substantially linear relationship therebetween and, thereby, the slope can be determined.
  • the stress ⁇ can be calculated through the use of the resulting slope and the elastic compliance constants S 11 , S 12 and S 13 of the crystal.
  • the calculation result of the tetragonal system having the Laue symmetry 4/mmm is different from that of the Laue symmetry 4/m.
  • the calculation result can be satisfactorily organized, and formulae suitable for the stress measurement are available.
  • the tetragonal system belonging to 4/m no formula suitable for the stress measurement is currently available. Consequently, according to the present invention, with respect to the “plane stress state” (general stress state of the specimen surface), the stress of the tetragonal system having only the Laue symmetry 4/mmm can be measured.
  • formula (15) shown in FIG. 8 representing the elastic compliance constant is substituted into formula (14) shown in FIG. 7 based on formula (20) shown in FIG. 9 representing the “plane stress state”.
  • the present forms of these formulae are not suitable for the stress measurement.
  • FIG. 15 is a diagram showing the symmetrical property of the tetragonal system having the Laue symmetry 4/mmm, viewed from the c-axis direction.
  • the symmetry viewed from the c-axis direction the symmetry relating to the present invention, is composed of a combination of a four-fold symmetry about the c-axis and one mirror symmetry.
  • An X-ray optical system is arranged in the location indicated by ⁇ .
  • the terms of sin 4 ⁇ cancel each other to become zero, and the terms of cos 4 ⁇ remain as a term of cos 4 ⁇ with no change.
  • the result of the calculation is formula (24) shown in FIG. 12 .
  • a horizontal line drawn above ⁇ L 33 (0°) refers to “an average value” of the above-described eight types of reflection.
  • the stress measurement can be performed through the use of these formulae as described below.
  • a strain ⁇ is calculated from the diffraction angle ⁇ and the diffraction angle ⁇ 0 (known) in a non-strain state.
  • the resulting strain ⁇ corresponds to an average value of ⁇ L 33 (0°).
  • ⁇ L 33 90°
  • FIG. 1 the horizontal axis indicates sin 2 ⁇
  • each measurement value is plotted, and an approximate linear line of the resulting graph is determined, for example, a regression line is determined by a least-squares method.
  • the value of ⁇ 11 + ⁇ 22 can be determined from the slop of the resulting linear line and the elastic compliance constants S 11 , S 12 , and S 13 .
  • the horizontal axis indicates V
  • each measurement value is plotted, and an approximate linear line of the resulting graph is determined.
  • the slop of the resulting linear line corresponds to ⁇ 11 ⁇ 22 .
  • the horizontal axis indicates V
  • each measurement value is plotted, and an approximate linear line of the resulting graph is determined.
  • the slop of the resulting linear line corresponds to 2 ⁇ 12 .
  • the Miller indices (hkl), the diffracted ray from which can be measured include those shown in FIG. 17 .
  • the direction of the normal to the crystal plane represented by the Miller indices (hkl) is the direction inclined at an angle of ⁇ from the direction of the normal to the specimen surface (that is, from the direction of the coordinate axis P 3 ) as shown in FIG. 18 .
  • the value of ⁇ with respect to each Miller indices is as shown in FIG. 17 .
  • the value of ⁇ , the lattice spacing d 0 in a non-strain state and the Bragg's angle ⁇ 0 corresponding thereto are also shown in FIG. 17 .
  • FIG. 18 is a diagram showing a state in which the L 3 -L 1 plane in the laboratory coordinate system is made parallel to the paper surface.
  • the direction of the normal to the crystal plane with the Miller indices (hkl) is the direction of the coordinate axis L 3 , and is inclined at an angle of ⁇ from the coordinate axis P 3 .
  • the crystal plane with the Miller indices (hkl) is parallel to the coordinate axis L 1 .
  • the X-ray 14 incident on the specimen 10 from the X-ray source 12 is diffracted at the crystal plane with the Miller indices (hkl), and the resulting diffracted X-ray 16 is detected with the X-ray detector 18 .
  • the X-ray source 12 and the X-ray detector 18 is arranged to be symmetrical with the normal to the crystal plane with the Miller indices (hkl).
  • the incident X-ray 14 and the crystal plane form an angle of ⁇
  • the diffracted X-ray 16 and the crystal plane form an angle of ⁇ as well.
  • the Bragg's angle ⁇ 0 in a non-strain state is thereby determined (known). Therefore, the X-ray source 12 and the X-ray detector 18 are adjusted within the range of a very small angle in the vicinity of this ⁇ 0 so that a diffraction angle ⁇ at which the intensity of the diffracted x-ray becomes a maximum can be found out.
  • This value ⁇ is taken as a measurement value.
  • a strain ⁇ can be calculated from the difference between this measurement value ⁇ and the Bragg's angle ⁇ 0 .
  • An X-ray optical system is set in an arbitrary plane including the coordinate axis P 3 (any plane can be selected because of the equi-biaxial stress state), and X-ray diffraction measurement is performed with respect to a plurality of ⁇ , diffracted rays from which can be detected, so that each diffraction angle ⁇ is measured. With respect to each ⁇ , a strain ⁇ is calculated from the diffraction angle ⁇ and the diffraction angle ⁇ 0 (known) in a non-strain state.
  • results are plotted while the horizontal axis indicates sin 2 ⁇ , and the vertical axis indicates ⁇ L 33 .
  • An approximate linear line of the resulting graph is determined, for example, a regression line is determined by a least-squares method.
  • the value of ⁇ can be determined from the slop of the resulting linear line and the elastic compliance constants S 11 , S 12 , and S 13 .

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